Pusat Penelitian dan Pengembangan Pendidikan Matematika Realistik Indonesia Sekolah Tinggi Keguruan dan Ilmu Pendidikan

Place Value on the Horizon

Understanding the number system has an essential connection with the history of number system itself. The development of number system in history has undergone many stages across culture and over time. It takes different times and different places that relate each other – the development in one place may affect the development in others for the future. In that case, it will create various kinds of symbol that uniquely depend on the condition in that era. Not only the history of number system which has passed through many ways to get the general symbol, but also the way of children understanding in number system. Children encounter the difficulty in learning the number system but by using their creative thinking, they finally find the solution to overcome the problem. The creative thinking will conduct them to find the suitable understanding of number system. They have learned it from the young ages with some phases of learning, starting from one by one counting strategy, grouping, knowing a notation,  adding and multiplying structure, and  unitizing to understanding the place value.

The history of number system began from Paleolithic era when it has been found the first numerical marks such as slashes, or tallies, carved onto cave walls or into bone, wood, or stone. One slash meant one object and ten reindeer meant ten tallies. Once, a problem was occurred when the system could not represent large numbers. After that, the Incas refined the system of Persia’s people, developing a cord which has held horizontally from which knotted strings were hung. It represented a value of single units, tens, and hundreds in which can be seen from the type of knots that has been used, the length of the cord, and the color and the position of the strings. This system implied a grouping of ten used by the Incas. It’s different from Sumerian; they used clay stones to represent a value of one: a ball meant ten; and a large cone meant sixty. It was implied that they used a grouping of six which was different with previous system.

While in Babylonian times, the first mathematical symbol has been found such as a nail shape meant units and a chevron shape meant tens. Some people in different cultures and times have a rather same system. Mayans used a bar which meant five and a dot meant unit. Egyptian had a special system also used lines which meant ones, a basket meant tens, a coiled rope meant hundreds, and a lotus flower meant thousands. Roman also represented the symbols such as C which meant one hundred, L meant fifty, X meant ten, V meant five, and I meant one. All systems are considered as an additive numeration system represented by symbol.

The notation symbol in nowadays is the result of the evolution of number systems. This idea came up with the multiplicative operation so that there is no symbol again needed to represent tens or hundreds. This was made in an easy way to grasp the mathematical concept quickly which the idea was different from the previous system.

It is necessary for children to understand the concept of number system but it takes time and also it’s gradually develop slowly for children to understand the concept. It can be seen from the young age of children in which they seemed to have no connection at all to quantity. They just drew pictures of the objects that had no attempt to describe or represent the amount of the object. The Similar case was happened in Jodi’s class, when Jodi asked her students; Noah, he only could count for 10 plus 10. One of her students said,” this is easy” by showing the block one by one, Gabrielle counted them: 10, 30, 40, 50, 60, 70, 80, 90, 100.  There are two mistakes from her counting; first, she missed 20 (no one to one correspondence – skipping the numbers), and also the binding block is only 5 not 10. It implied that Noah and Gabrielle still lack of understanding in one to one correspondence.

If the children have understood one to one correspondence, children would begin to construct the idea of one to one correspondence by representing the quantity of pictograph representation. For example there are two cases from Jack and Susie that is served in the book. Jack had a bag with seven teddy bears and he drew every bear in the bag, and Susie had a bag with fifteen Unifix cubes. These bring Jodi to pose a question how children can represent the larger amounts of teddy bears/bags.

Later children would take iconic representation because they had understood of cardinality in which one symbol can represent the whole amount. For example, Raquel made a poster with buttons that represented twenty-eight children. She wrote counting numerals next to each button. It meant that she represented her counting action but not the cardinality.

Many cases happened in different solution from the student’s works. From Raquel’s work, he does not represent the result of counting but only the counting itself. On the other hand, Bryce wrote only numerals and also represented a counting. His work proved that children start to construct a new idea and the process or evidence of the path is still important.

Next step is grouping, this thing happened when the children felt tired of counting one by one and attempting to find a new way to count by grouping the amount. From the book we can see that the movement from grouping to notation to additive structure is not really hard to do. LeeAnne’s and Bill’s representation are the beginning of an additive system which used color group of six and ten sequentially with the notation. It seemed they pick the number of group related to the number which familiar for them.

Besides all stages that we have described, there is one important thing to determine the way of learning in classroom namely context. Working with context really help children to communicate to the teacher and also for the teacher,  it’s useful to guide children’s thinking to a right understanding of place value easily. We could see the result of Jodi’s context: Taking Inventory. In this occasion, Jodi made herself confused with the condition of their class and asked to help to all children to make a sign in all stuffs. Jodi might want children to do the learning activity seriously and planted a sense of responsibility in children’s mind in order to make children having a big desire to study in that time.

In supporting the beginning ability to represent groups of ten symbolically, Jodi took an inventory of the classroom books and supply rubber bands. Jodi indicated children to find “How many?” as a context, such as: how many packs of ten books, how many loose books, and how many books all together. It was also shown in the class of Jodi, when Isa and Eli were asked by Jodi to make a report of bandages. Jodi began with the question if there were 60 and 80 bandages, how many packs missing for each bandage. First, Eli took paper bag with 80 bandages on it and tried to predict if there were only 79 bandages in the bag. While Isa predicted another bag and told her if it was only 50 bandages in that bag. But Jodi emphasized once again about her question. Jodi asked them for 60th bag, then the 80th bag. In that step, Eli said if 20 equals to 2 and 30 equals to 3 packs. From that activity, we can see how patience Jodi in guiding her students so that Isa and Eli could find their own answers.

From here, we can conclude that not only history aspect of understanding the number system that emerge gradually refinement, but also the development of children thinking to perceive number systems. There are many steps which are needed to be through passed with struggling by the children in order to construct an important mathematical idea of number systems. All of these phases practice the children for their capacity in doing mathematical problems.


By Ekasatya Aldila Afriansyah

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